18.1 What are Hurdle Models and When are they Useful?
Hurdle models are a specialized statistical tool designed to handle data with a preponderance of zero values, which often defy the assumptions of conventional distributions like the normal or Poisson. These models are particularly valuable when the data-generating process naturally consists of two distinct stages:
Process 1: The Hurdle - Zero or Non-Zero? This stage acts as the gatekeeper, determining whether the underlying data-generating mechanism is “active” or “dormant.” It essentially answers the question: Is the outcome zero or non-zero?
Process 2: Modeling the Non-Zeros. Contingent upon the outcome of Process 1, if the result is non-zero, this second stage steps in to model the specific value it assumes. The choice of distribution for this modeling phase hinges on the nature of the data and could encompass lognormal, gamma, Poisson, or negative binomial distributions.
Consider the scenario of analyzing user engagement with a new software feature. Some users might never activate the feature, perhaps due to lack of awareness or need. Others who find it valuable might use it to varying extents. A hurdle model effectively captures both the probability of a user engaging with the feature at all (Process 1 - crossing the ‘hurdle’), and the extent of their engagement if they do (Process 2 - the usage intensity).
Mathematical Representation for a Hurdle Model with a Log Normal Component
Let’s delve into the mathematical underpinnings, keeping it as intuitive as possible.
Step 1: The Zero-Inflation Component
Let \(Z_i\) be a binary indicator variable for whether observation \(i\) is zero.
\(Z_i \sim Bernoulli(\theta_i)\)
where \(\theta_i = logit^{-1}(\alpha_{zero} + \tilde{X}_i \beta_{zero} + \tau_{zero} T_i)\)
\(\tilde{X}_i\) is the standardized design matrix (predictor variables) for observation \(i\)
\(T_i\) is the treatment indicator (1 for treatment, 0 for control).
Let’s bring these concepts to life with a practical example. Imagine you’re part of an online video content company aiming to assess the impact of a novel marketing strategy on video watch time. You notice that certain videos garner substantial watch time, while others remain unwatched. To rigorously test this new strategy, you randomly assign videos to either a treatment or control group. Here’s how we might simulate such data:
library(ggplot2)ggplot(fake_data, aes(x = watch_time)) +geom_histogram(aes(y =after_stat(density)), bins =30, fill ="lightblue", color ="black") +labs(title ="Distribution of Watch Time", x ="Watch Time", y ="Density") +theme_minimal()
In this simulated data, the average treatment effect leads to an increase in watch time of approximately 30 hours. Additionally, the probability of having zero hours of watch time decreases by -3 percentage points.
18.3 Pitfalls of Naive OLS Regression
Since treatment assignment is random in our simulation, it might be tempting to use a simple linear regression:
However, this approach overestimates the true treatment effect. The point estimate and confidence intervals are significantly higher than the actual impact.
Another common approach is to log-transform the outcome variable and then apply OLS:
# Run the OLS regression with log(watch_time + 1) as the dependent variablelm2 <-lm(data = fake_data, log(watch_time +1) ~ genre + popular_channel + length + treatment) |> broom::tidy(conf.int =TRUE)lm2
Coordinate system already present. Adding new coordinate system, which will
replace the existing one.
ppc_plot # Display the plot
The posterior predictive check helps us gauge whether the model adequately captures the characteristics of the observed data.
Finally, let’s extract the key estimates:
# Get ATE point estimate and credible intervalate <- model$pointEstimate("ATE")ci_statement <- model$credibleInterval("ATE")# Print resultscat("The mean of the posterior distribution of the average treatment effect is", round(ate))
The mean of the posterior distribution of the average treatment effect is 32
ci_statement
Given the data, we estimate that there is a 95% probability that the ATE is between 26.52 and 37.9.
# Get ATE point estimate and credible intervaltau_prob_zero <- model$pointEstimate("tau_prob_zero") ci_statement <- model$credibleInterval("tau_prob_zero")# Print resultscat("The mean of the posterior distribution of the effect on the probability of a zero is", round(tau_prob_zero,4))
The mean of the posterior distribution of the effect on the probability of a zero is -0.0457
ci_statement
Given the data, we estimate that there is a 95% probability that the tau_prob_zero is between -0.09 and 0.
model$calcProb(effect_type ="ATE", a =29)
Given the data, we estimate that the probability that the ATE is more than 29 is 86%.
The Bayesian hurdle model, as implemented with {imt}, provides a more nuanced and accurate assessment of the treatment effect in the presence of excessive zeros, outperforming naive OLS approaches.
18.5 Hurdle vs. Zero-Inflated Models: Choosing the Right Tool for the Job
While hurdle models excel at handling data with excess zeros, it’s crucial to distinguish them from another class of models designed for similar scenarios: zero-inflated models. Both tackle the challenge of zero inflation, but they do so with subtle yet important differences.
Conceptual Differences
Hurdle Models: Assume a single process generates both zeros and non-zeros. The “hurdle” represents a threshold that must be crossed before any non-zero outcome can occur. Think of it as a binary decision: either the outcome is zero, or it’s something positive that we then model with a suitable distribution.
Zero-Inflated Models: Assume two distinct processes are at play. One process generates only zeros (the “structural zeros”), while another process generates both zeros and non-zeros (the “count process”). This allows for the possibility of “excess zeros” beyond what would be expected from the count process alone.
When to Use Each
Hurdle Models: Ideal when there’s a clear conceptual hurdle or threshold that needs to be overcome before a non-zero outcome can happen. For instance, in our video watch time example, users need to decide to watch a video at all before any watch time can be recorded.
Zero-Inflated Models: More suitable when you suspect there are two fundamentally different types of zeros in your data. For example, in a survey about alcohol consumption, some respondents might be teetotalers (structural zeros), while others might simply not have consumed alcohol during the survey period (zeros from the count process).
The choice between a hurdle model and a zero-inflated model hinges on your understanding of the data-generating process and the research question at hand. If you believe there’s a single process with a clear hurdle, a hurdle model is the way to go. If you suspect multiple processes leading to zeros, a zero-inflated model might be more appropriate.
In our video watch time example, we opted for a hurdle model because the decision to watch a video (or not) seemed like a natural hurdle. However, if we had reason to believe that some videos were inherently unappealing and would never be watched by anyone (structural zeros), a zero-inflated model might have been worth considering.
The key takeaway is that both hurdle and zero-inflated models offer powerful ways to handle excess zeros, but their underlying assumptions and interpretations differ.
By carefully considering the nature of your data and your research goals, you can choose the model that best suits your needs and unlocks valuable insights hidden within the zeros.
Learn more
Team (2024) Stan User’s Guide: Finite Mixtures and Zero-inflated Models.
Heiss (2022): A guide to modeling outcomes that have lots of zeros with Bayesian hurdle lognormal and hurdle Gaussian regression models.
---title: "Hurdle Models"share: permalink: "https://book.martinez.fyi/hurdle.html" description: "Business Data Science: Hurdle Models" linkedin: true email: true mastodon: true---<img src="img/hurdle.jpg" align="right" height="280" alt="Hurdle Models" />## What are Hurdle Models and When are they Useful?Hurdle models are a specialized statistical tool designed to handle data with apreponderance of zero values, which often defy the assumptions of conventionaldistributions like the normal or Poisson. These models are particularly valuablewhen the data-generating process naturally consists of two distinct stages:1. **Process 1: The Hurdle - Zero or Non-Zero?** This stage acts as the gatekeeper, determining whether the underlying data-generating mechanism is "active" or "dormant." It essentially answers the question: Is the outcome zero or non-zero?2. **Process 2: Modeling the Non-Zeros.** Contingent upon the outcome of Process 1, if the result is non-zero, this second stage steps in to model the specific value it assumes. The choice of distribution for this modeling phase hinges on the nature of the data and could encompass lognormal, gamma, Poisson, or negative binomial distributions.Consider the scenario of analyzing user engagement with a new software feature.Some users might never activate the feature, perhaps due to lack of awareness orneed. Others who find it valuable might use it to varying extents. A hurdlemodel effectively captures both the probability of a user engaging with thefeature at all (Process 1 - crossing the 'hurdle'), and the extent of theirengagement if they do (Process 2 - the usage intensity).### Mathematical Representation for a Hurdle Model with a Log Normal ComponentLet's delve into the mathematical underpinnings, keeping it as intuitive as possible.**Step 1: The Zero-Inflation Component*** Let $Z_i$ be a binary indicator variable for whether observation $i$ is zero.* $Z_i \sim Bernoulli(\theta_i)$ * where $\theta_i = logit^{-1}(\alpha_{zero} + \tilde{X}_i \beta_{zero} + \tau_{zero} T_i)$ * $\tilde{X}_i$ is the standardized design matrix (predictor variables) for observation $i$* $T_i$ is the treatment indicator (1 for treatment, 0 for control).* Priors: * $\alpha_{zero} \sim Normal(\mu_{\alpha_{logit}}, \sigma_{\alpha_{logit}})$ * $\beta_{zero} \sim Normal(\mu_{\beta_{logit}}, \sigma_{\beta_{logit}})$ * $\tau_{zero} \sim Normal(\mu_{\tau_{logit}}, \sigma_{\tau_{logit}})$**Log-Normal Component*** Let $Y_i$ be the outcome variable.* If $Z_i = 0$ (i.e., the outcome is zero), then $Y_i = 0$.* If $Z_i = 1$ (i.e., the outcome is positive), then: * $log(Y_i) \sim Normal(\mu_i, \sigma_{lnorm}^2)$ * where $\mu_i = \alpha_{lnorm} + \tilde{X}_i \beta_{lnorm} + \tau_{lnorm} T_i$* Priors: * $\alpha_{lnorm} \sim Normal(\mu_{log(y)}, 1)$ * $\beta_{lnorm} \sim Normal(0, 0.5)$ * $\tau_{lnorm} \sim Normal(\mu_{\tau}, \sigma_{\tau})$ * $\sigma_{lnorm} \sim Normal(0, 0.5)$ **Combined Model*** The overall likelihood for observation $i$ is:* $P(Y_i = 0) = \theta_i$* $P(Y_i > 0) = (1 - \theta_i) \times \frac{1}{Y_i \sigma_{lnorm} \sqrt{2\pi}} exp \left( -\frac{(log(Y_i) - \mu_i)^2}{2 \sigma_{lnorm}^2} \right)$## Simulating Data for a Hurdle ModelLet's bring these concepts to life with a practical example. Imagine you're partof an online video content company aiming to assess the impact of a novelmarketing strategy on video watch time. You notice that certain videos garnersubstantial watch time, while others remain unwatched. To rigorously test thisnew strategy, you randomly assign videos to either a treatment or control group.Here's how we might simulate such data:```{r message=FALSE, warning=FALSE}library(dplyr)set.seed(123)n <- 3000# Simulate covariates (unchanged)fake_data <- data.frame( genre = sample(c("Comedy", "Education", "Music"), n, replace = TRUE), length = stats::rnorm(n, mean = 10, sd = 3), popular_channel = stats::rbinom(n, 1, 0.2))# Treatment indicator (unchanged)fake_data$treatment <- stats::rbinom(n, 1, 0.5)# Model parameters (coefficients) - unchangedbeta_zero <- c(0.5, -0.2, 1)beta_mean <- c(2, 0.1, 0.5)# Modified treatment effect for zero probability# We want P(zero | treated) = P(zero | control) - 0.05# Assuming P(zero | control) = 0.3 p_zero_control <- 0.3p_zero_treated <- p_zero_control - 0.05treatment_effect_zero_prob <- qlogis(p_zero_treated) - qlogis(p_zero_control)# Treatment effect for watch time treatment_effect_watch_time <- 2# Linear predictors (with modified treatment effect)zero_prob_logit <- beta_zero[1] + ifelse(fake_data$genre == "Education", beta_zero[2], 0) + beta_zero[3] * fake_data$popular_channel + treatment_effect_zero_prob * fake_data$treatmentlog_normal_mean <- beta_mean[1] + beta_mean[2] * fake_data$length + beta_mean[3] * fake_data$popular_channel + treatment_effect_watch_time * fake_data$treatment# Generate potential outcomesfake_data <- within(fake_data, { zero_prob_control <- stats::plogis(zero_prob_logit - treatment_effect_zero_prob * treatment) zero_prob_treated <- stats::plogis(zero_prob_logit) watch_time_control <- ifelse(stats::rbinom(n, 1, 1 - zero_prob_control) == 1, stats::rlnorm(n, log_normal_mean - treatment_effect_watch_time * treatment, 0.5), 0) watch_time_treated <- ifelse(stats::rbinom(n, 1, 1 - zero_prob_treated) == 1, stats::rlnorm(n, log_normal_mean, 0.5), 0) # Calculate treatment effects tau_watch_time <- watch_time_treated - watch_time_control tau_zero_prob <- zero_prob_treated - zero_prob_control # Observed outcome based on treatment assignment zero_prob <- ifelse(treatment == 1, zero_prob_treated, zero_prob_control) watch_time <- ifelse(treatment == 1, watch_time_treated, watch_time_control)}) |> dplyr::mutate(treatment = as.logical(treatment))dplyr::glimpse(fake_data)``````{r}library(ggplot2)ggplot(fake_data, aes(x = watch_time)) +geom_histogram(aes(y =after_stat(density)), bins =30, fill ="lightblue", color ="black") +labs(title ="Distribution of Watch Time", x ="Watch Time", y ="Density") +theme_minimal()```In this simulated data, the average treatment effect leads to an increase inwatch time of approximately `r round(mean(fake_data$tau_watch_time))` hours.Additionally, the probability of having zero hours of watch time decreases by`r round(100*mean(fake_data$tau_zero_prob))` percentage points.## Pitfalls of Naive OLS RegressionSince treatment assignment is random in our simulation, it might betempting to use a simple linear regression:```{r}lm1 <-lm(data = fake_data, watch_time ~ genre + popular_channel + length + treatment) |> broom::tidy(conf.int =TRUE)lm1```However, this approach overestimates the true treatment effect. Thepoint estimate and confidence intervals are significantly higher thanthe actual impact.Another common approach is to log-transform the outcome variable andthen apply OLS:```{r}# Run the OLS regression with log(watch_time + 1) as the dependent variablelm2 <-lm(data = fake_data, log(watch_time +1) ~ genre + popular_channel + length + treatment) |> broom::tidy(conf.int =TRUE)lm2mean_watch_time_control <-mean(fake_data$watch_time[fake_data$treatment ==FALSE]) point_estimate <- mean_watch_time_control*(1+lm2$estimate[6])lb <- mean_watch_time_control*(1+lm2$conf.low[6])ub <- mean_watch_time_control*(1+lm2$conf.high[6])```In this case, the point estimate (`r round(point_estimate)`) and theconfidence interval ([`r round(lb)`, `r round(ub)`]) underestimate thetrue effect.## Bayesian Hurdle Model Using {imt}<img src="https://raw.githubusercontent.com/google/imt/refs/heads/main/man/figures/logo.png" align="right" height="138" alt="logo of the imt package" />Let's use the [{imt}](https://github.com/google/imt) package to fit a Bayesianhurdle model:```{r fit, message=FALSE, results = "hide"}library(imt)# Create the hurdleLogNormal objectmodel <- hurdleLogNormal$new( data = fake_data, y = "watch_time", x = c("length", "popular_channel", "genre"), treatment = "treatment", tau_mean_logit = 0, tau_sd_logit = 0.5, mean_tau = 0, sigma_tau = 0.035)```### Plot PriorsBefore diving into the posterior analysis, let's visualize the priors we've setfor our model:```{r}prior_plots <- model$plotPrior(bins =1000,xlim_ate =c(-500, 500),xlim_tau =c(-10, 10))# To display the plots:prior_plots$ate_priorprior_plots$tau_prior```These plots provide a visual representation of our prior beliefs about thetreatment effects before observing the data.### Posterior AnalysisNow, let's examine the posterior distribution and assess the model's fit:```{r}ppc_plot <- model$posteriorPredictiveCheck(n =50, xlim =c(0, 500))ppc_plot # Display the plot```The posterior predictive check helps us gauge whether the model adequatelycaptures the characteristics of the observed data.Finally, let's extract the key estimates:```{r}# Get ATE point estimate and credible intervalate <- model$pointEstimate("ATE")ci_statement <- model$credibleInterval("ATE")# Print resultscat("The mean of the posterior distribution of the average treatment effect is", round(ate))ci_statement``````{r}# Get ATE point estimate and credible intervaltau_prob_zero <- model$pointEstimate("tau_prob_zero") ci_statement <- model$credibleInterval("tau_prob_zero")# Print resultscat("The mean of the posterior distribution of the effect on the probability of a zero is", round(tau_prob_zero,4))ci_statement``````{r}model$calcProb(effect_type ="ATE", a =29)```The Bayesian hurdle model, as implemented with {imt}, provides a more nuancedand accurate assessment of the treatment effect in the presence of excessivezeros, outperforming naive OLS approaches.## Hurdle vs. Zero-Inflated Models: Choosing the Right Tool for the JobWhile hurdle models excel at handling data with excess zeros, it's crucial todistinguish them from another class of models designed for similar scenarios:zero-inflated models. Both tackle the challenge of zero inflation, but they doso with subtle yet important differences.### Conceptual Differences - **Hurdle Models:** Assume a single process generates both zeros and non-zeros. The "hurdle" represents a threshold that must be crossed before any non-zero outcome can occur. Think of it as a binary decision: either the outcome is zero, or it's something positive that we then model with a suitable distribution. - **Zero-Inflated Models:** Assume two distinct processes are at play. One process generates only zeros (the "structural zeros"), while another process generates both zeros and non-zeros (the "count process"). This allows for the possibility of "excess zeros" beyond what would be expected from the count process alone.### When to Use Each - **Hurdle Models:** Ideal when there's a clear conceptual hurdle or threshold that needs to be overcome before a non-zero outcome can happen. For instance, in our video watch time example, users need to decide to watch a video at all before any watch time can be recorded. - **Zero-Inflated Models:** More suitable when you suspect there are two fundamentally different types of zeros in your data. For example, in a survey about alcohol consumption, some respondents might be teetotalers (structural zeros), while others might simply not have consumed alcohol during the survey period (zeros from the count process).The choice between a hurdle model and a zero-inflated model hinges on yourunderstanding of the data-generating process and the research question at hand.If you believe there's a single process with a clear hurdle, a hurdle model isthe way to go. If you suspect multiple processes leading to zeros, azero-inflated model might be more appropriate.In our video watch time example, we opted for a hurdle model because thedecision to watch a video (or not) seemed like a natural hurdle. However, if wehad reason to believe that some videos were inherently unappealing and wouldnever be watched by anyone (structural zeros), a zero-inflated model might havebeen worth considering.The key takeaway is that both hurdle and zero-inflated models offer powerfulways to handle excess zeros, but their underlying assumptions andinterpretations differ.By carefully considering the nature of your data and your research goals, youcan choose the model that best suits your needs and unlocks valuable insightshidden within the zeros. ::: {.callout-tip}## Learn more - @mc-stan2024finite Stan User's Guide: Finite Mixtures and Zero-inflated Models. - @heiss2022hurdle: A guide to modeling outcomes that have lots of zeros with Bayesian hurdle lognormal and hurdle Gaussian regression models.:::
